#8174408

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,7)(4,8)(5,6)(9,10,15,19)(11,16,21,24)(12,13,17,22)(14,23,20,18)(26,28)(30,38)(32,33)(35,37);(1,3)(2,4)(5,8)(6,7)(9,11,15,21)(10,16,19,24)(12,20,17,14)(13,18,22,23)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,12,15,17)(10,13,19,22)(11,14,21,20)(16,23,24,18)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(2,6)(3,8)(9,12)(10,13)(11,20)(14,21)(15,17)(16,18)(19,22)(23,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,18)(12,19)(14,16)(15,22)(20,24)(21,23)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(1,4)(2,3)(5,7)(6,8)(9,14,15,20)(10,18,19,23)(11,12,21,17)(13,16,22,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081604);